Institute of Mathematics

Abstract:  We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $(\rm SBaw)$, $(\rm SBab)$, $(\rm SBw)$ and $(\rm SBb)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $(\rm SBab)$, then $S\oplus T$ has the property $(\rm SBab)$ if and only if $\sigma_ SBF_+^-(S\oplus T)=\sigma_ SBF_+^-(S)\cup\sigma_ SBF_+^-(T)$, where $\sigma_ SBF_+^-(T)$ is the upper semi-B-Weyl spectrum of $T$. We obtain analogous preservation results for the properties $( SBaw)$, $(\rm SBb)$ and $(\rm SBw)$ with extra assumptions.